Damped vibration analysis of lumped mass on bidirectionally graded beam rested on torsion spring hinges

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This article is concerned with the vibration control of a bidirectionally graded beam supported on torsion spring hinges using a lumped mass damper. The conventional mixing rule is used to model the material gradation in the thickness direction. The material gradation in the axial direction is modeled by an exponential function. The spectral Ritz method is employed to minimize the total potential energy and calculate the fundamental angular frequency of free vibration and the corresponding mode shape. The characteristic vibration equation of the system is obtained by calculating the determinant of the Hessian of the total potential energy. For the first time, the modified Taylor basis is introduced, which eliminates the drawbacks and the necessity of using auxiliary functions in the spectral Ritz method. For this purpose, the modified Taylor basis is calculated by satisfying the boundary conditions and the natural conditions at the ends of the beam. The effects of the dimensionless rotational stiffness at the ends of the beam, the material gradations in the axial and transverse directions, the amount and position of the lumped mass on the fundamental angular frequency of the free vibration are investigated.